Draft version February 11, 2013
Preprint typeset using LATEX style emulateapj v. 08/22/09
CORONAL ALFVÉN SPEED DETERMINATION: CONSISTENCY BETWEEN SEISMOLOGY USING
AIA/SDO TRANSVERSE LOOP OSCILLATIONS AND MAGNETIC EXTRAPOLATION
E. Verwichte1,2 ,
T. Van Doorsselaere2 , C. Foullon1 , and R.S. White1
Draft version February 11, 2013
ABSTRACT
Two transversely oscillating coronal loops are investigated in detail during a flare on the 6th September 2011 using data from the Atmospheric Imaging Assembly (AIA) on board the Solar Dynamics
Observatory (SDO). We compare two independent methods to determine the Alfvén speed inside these
loops. Through the period of oscillation and loop length information about the Alfvén speed inside
each loop is deduced seismologically. This is compared with the Alfvén speed profiles deduced from
magnetic extrapolation and spectral methods using AIA bandpass. We find that for both loops the
two methods are consistent. Also, we find that the average Alfvén speed based on loop travel time
is not necessarily a good measure to compare with the seismological result, which explains earlier
reported discrepancies. Instead, the effect of density and magnetic stratification on the wave mode
has to be taken into account. We discuss the implications of combining seismological, extrapolation
and spectral methods in deducing the physical properties of coronal loops.
Subject headings: MHD, Sun: corona, magnetic fields, oscillations
1. INTRODUCTION
The solar corona and the structures therein, such as
loops, support magnetohydrodynamic (MHD) waves of
various kinds (Deforest & Gurman 1998; Aschwanden
et al. 1999; Berghmans & Clette 1999; Kliem et al. 2002;
Verwichte et al. 2005; Tomczyk et al. 2007). Transverse waves, and in particular transverse loop oscillations (TLOs) have received much attention because they
are manifestations of waves supported by the magnetic
field and plasma structuring of the corona (Aschwanden
et al. 1999; Nakariakov et al. 1999; Verwichte et al. 2004;
Tomczyk et al. 2007; McIntosh et al. 2011). The synoptic, full-disk nature of AIA on board SDO (Lemen et al.
2012) observations confirm that they are ubiquitous and
are present in many eruptive events (White & Verwichte
2012). TLOs occur in loops of all sizes and temperatures,
have periods spanning a few to tens of minutes (e.g. Aschwanden et al. 2002; Verwichte et al. 2010; White &
Verwichte 2012; White et al. 2012).
The comparison of the observed wave quantities with
MHD wave theory provides a seismological route to determining local physical parameters that are difficult to
measure directly (e.g. Verwichte et al. 2006b; Arregui
et al. 2007; Goossens et al. 2008). Nakariakov & Ofman
(2001) demonstrated that by measuring the period and
wavelength of the oscillation an estimate of the loop’s
Alfvén speed and local magnetic field strength can be
obtained. Knowledge of those physical parameters is
important for modelling dynamics in the solar corona
such as solar flares and CMEs. Various authors have reported phase speeds typically of the order of 1000 kms−1 .
The determination of the Alfvén speed from the observed
phase speed lies at the heart of the seismological method
Electronic address: Erwin.Verwichte@warwick.ac.uk
1 Centre for Fusion, Space and Astrophysics, Department of
Physics, University of Warwick, Coventry CV4 7AL, UK
2 Centre for Plasma Astrophysics, Department of Mathematics,
Katholieke Universiteit Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium
and employs the wave theory for kink modes. The basic theory by Edwin & Roberts (1983) applies to internally uniform loops. However, to improve accuracy, effects such as density and magnetic stratification can be
incorporated (Andries et al. 2005, 2009). In all versions
of the wave theory, knowledge of the density contrast
between the loop and the external corona is required to
find the Alfvén speed. However, the exact value of the
density contrast is difficult to measure directly (e.g. Aschwanden et al. 2003; Schmelz et al. 2003; Terzo & Reale
2010).
Aschwanden & Schrijver (2011) were the first to compare a seismologically determined magnetic field with a
potential magnetic extrapolation. The seismologically
determined magnetic field strength had been extracted
using the density measured using a DEM method applied to the AIA bandpasses (Aschwanden et al. 2011;
Hannah & Kontar 2012). They found that it the average
extrapolated magnetic field strength in the loop exceeded
the seismologically determined value by a factor of three.
We wish to establish the physical reasons for such a discrepancy. Because the loop’s magnetic field strength is
derived from the Alfvén speed by using the loop density
(deduced from spectral measurement or from assumption), the Alfvén speed is the true seismologically determined physical loop quantity. We shall therefore determine the Alfvén speed from a new TLO event using
AIA data and compare it with the Alfvén speed obtained
by spectral and magnetic extrapolation methods. This
has become feasible because of the superior quality of
the AIA instrument and the multiple viewpoints made
possible by the Extreme UltraViolet Imager (EUVI) on
board STEREO (Howard et al. 2008; Wuelser et al. 2004;
Verwichte et al. 2009).
The paper is organised as follows. In Sect. 2, we
present the wave analysis two TLOs seen by AIA. In Sect.
3 the Alfvén speed using the ’direct’ methods of magnetic
extrapolation and DEM inversion is calculated. In Sect.
4 the role of stratification is examined in determining the
2
E. Verwichte et al.
Fig. 2.— time-distance image of full intensity for the path at
s=0.5L. Time is counted in minutes from reference time 22:00 UT
on 6 Sep 2011. Middle: Filtered time-distance image with the loop
position of loop #1 of Event A indicated. Bottom: displacement
time series ξ(t). The dashed line is a fitted damped sinusoidal
curve. The dotted line indicated the start time used for the fit.
Fig. 1.— Detailed view of the loops at 22:15:30 UT in AIA/SDO
171Å (upper) and in EUVI/STEREO-A 171Å (lower).
equivalent Alfvén speed to compare with wave observations. In Sect. 5, the results from the comparison are
used to determine the loop cross-section profile. Finally,
in Sect 6 we discuss our findings, explore its potential
and limitations.
2. TRANSVERSE LOOP OSCILLATION EVENT
We examine two loops in NOAA active region 11283
on the 6th of September 2011, which exhibit transverse
oscillations of periods around 2-3 minutes in response
to a GOES class X2.1 flare inside the region starting
at 22:12 UT and peaking at 22:20 UT. The location of
these loops as seen by AIA/SDO in the 171Å bandpass
in shown in Fig. 1. Following the technique detailed in
Verwichte et al. (2010) and White & Verwichte (2012)
the loops are compared with their view in images from
EUVI/STEREO-A 171Å at 22:15 UT to determine the
general 3d geometry (see Fig. 1). STEREO-A is ahead
of Earth by 103◦ in longitude. To the projected loop
path a third coordinate is added under the assumption
that the whole loop lies within a plane. The inclination
of that plane with respect to the photospheric normal, θ,
is now the only free parameter. The model loop is then
transformed to the second viewpoint and θ is adjusted
visually to best match the loop seen there. The threedimensional loop geometry gives loop lengths of 213 Mm
and 188 Mm, respectively. The loop geometry parameters are listed in Table 1.
We employ the tried and tested analysis technique for
transverse oscillations that has been perfected by several
studies (Verwichte et al. 2004; Van Doorsselaere et al.
2007; Verwichte et al. 2009, 2010; White & Verwichte
2012). A rectangular box region of interest (data cut)
is chosen that runs across the loop top in the direction aligned with the projected displacement of a hypothetical horizontally polarised TLO. The AIA time
series are then interpolated at the ROI locations. For
each time, we average the data over the 11-pixel width
of the box to increase the signal-to-noise ratio. Thus,
the dataset is reduced to a two-dimensional time-space
image set containing the spatial average intensity along
the ROI as a function of time. A TLO would appear
as a periodic displacement of the loop location. To obtain the loop axis displacement, we fit a Gaussian shape
(Carcedo et al. 2003) superimposed on a linear trend.
The width of the fitted Gaussian profile is taken to be
a good measure of error on the position. Finally, from
the position a trend is subtracted and the loop’s oscillation displacement time series, ξ(t), is obtained, which is
characterised by fitting a damped cosine function of the
form ξ(t) = ξ0 exp[−(t−t0 )/τ ] cos[2π(t−t0 )/P + φ] using a Levenburg-Marquardt least squares fitting method
Markwardt (2009). The fitting parameters are the displacement amplitude, ξ0 , oscillation period, P , damping
time, τ , phase φ and reference time, t0 . The oscillation
parameters are listed in Table 1.
The phase speed, Vph , is calculated as the ratio of the
wavelength (for fundamental standing mode λ = 2L),
over the period, where we allow for an error of typically
10% in L. For the two loops, we find phase speeds in the
Coronal Alfvén speed determination
range 2500-2600 kms−1 . By interpreting the TLO as a
Alfvénic kink mode in a thin, cylindrical loop uniform in
the longitudinal direction and in the zero plasma-β limit,
the phase speed is equal to the kink speed, CK , given by
(Edwin & Roberts 1983):
r
2
CK ≈
VA ,
(1)
1 + ζ −1
where ζ = ρ0i /ρ0e is the ratio of internal to external
densities and VA is the Alfvén speed in the loop. We can
thus calculate from the phase speed the value range of
the loop Alfvén speed,
r
1 + ζ −1
VA,s = Vph
,
(2)
2
considering the two extreme values of ζ, namely unity
and infinity for an overdense loop. The exact value of the
density contrast is difficult to measure (e.g. Aschwanden
et al. 2003; Schmelz et al. 2003; Terzo & Reale 2010).
Expression (2) assumes that the loop is uniform with the
internal Alfvén speed equal to VA,s all along the loop.
However, it is expected that the loops are stratified in
density and magnetic field strength and that the Alfvén
speed varies along the loop. Aschwanden & Schrijver
(2011) equate VA,s with the average speed in the loop for
to the same travel time.
3. COMPARISON BETWEEN SEISMOLOGICAL AND
DIRECT DETERMINATION OF ALFVÉN SPEED
The Alfvén speed VA,s can be exploited seismologically
to determine the loop magnetic field strength (Nakariakov & Ofman 2001). However, in order to do so the
value of the loop density is required. Often in absence
of spectroscopic information of the loop density, and
employing the argument that the density appears only
weakly (through a square root) in the Alfvén speed, a
plausible value of the electron number density ne =1015
m−3 had been assumed (Verwichte et al. 2004). For this
density, we find values for the magnetic field strength for
each loop in the ranges 25-46 G and 25-48 G, respectively. If we assume a density one order of magnitude
smaller, we find values of 8-15 G for both loops instead.
We shall attempt to test the consistency of the seismological method by estimating the magnetic field and loop
density independently. We can then compare the average
Alfvén speeds from both methods.
First, the magnetic field is determined using the Potential Field Source Surface (PFSS) potential field extrapolation tool (Schrijver 2001; Schrijver & De Rosa 2003),
which produces a data cube of all three components of
magnetic field in spherical coordinates. Figure 3 shows
the potential field extrapolation of the active region. The
projected loop paths are indicated in red. The magnetic
field is interpolated for the three-dimensional loop paths.
We can see that the paths approximately align with the
potential field, but not fully. Misalignments with an angle between 20 and 40 degrees between potential-field
extrapolations and loops have been reported in the past
(Sandman et al. 2009). The average ratio of parallel to
total magnetic field strength is 60%. Importantly, for
our study only the field strength is required, which is expected to be less sensitive. The top panels in Figures 4
and 5 shows the magnetic field strength along the loop.
3
TABLE 1
Physical loop quantities
Quantities
Loop #1
Loop #2
Loop length L
188 ± 20 Mm
160 ± 20 Mm
Loop inclination angle θ
27◦ N
25◦ N
Loop height h
64 Mm
49 Mm
Footpoint separation ∆α
7.1◦
7.6◦
Loop geomety
Oscillation parameters
Oscillation period P
150 ± 5, s
122 ± 6, s
Damping time τ
216 ± 60, s
348 ± 400, s
Displacement amplitude ξ0
6.9 ± 1 Mm
1.9 ± 1 Mm
Phase φ
340 ± 10◦
299 ± 30◦
Reference time t0
22 : 04 UT
22 : 04 UT
Mode number n
1
1
Wavelength λ
380 ± 40 Mm
320 ± 30 Mm
Phase speed Vph
2510 ± 400 km/s 2620 ± 400 km/s
Seis. Alfvén speed VA,s
1780−2510 km/s 1860−2620 km/s
Extrapolated quantities
Av. magn. field <B>
26 G
41 G
Weig. magn. field <BW>
19 G
32 G
Footpoint density ne,fp
0.7 1015 m−3
0.7 1015 m−3
139 Mm
Density scaleheight H
65 Mm
Loop temperature T0
0.79 MK
0.79 MK
Loop minor radius a
0.95 Mm
0.85 Mm
Alfvén speed scaleheight Λ
107 Mm
98 Mm
loop top Alfvén speed VA,top
1470 km/s
1640 km/s
Av. Alfvén speed <VA>
2960 km/s
3430 km/s
Weig. Alfvén speed <VA,W> 2260 km/s
2700 km/s
Dir. Alfvén speed VA,d
2130 km/s
2480 km/s
0.4 (0.1−0.9)
0.8 (0.3−1)
Loop cross−sectional profile
Inverse density contrast ζ −1
Transition layer thickness ℓ/a 1.1 (0.6−2)
1.9 (0−2)
We estimate the density of the loops from the AIA
bandpasses following a method outlined in Aschwanden
& Schrijver (2011). However, as the loops are only seen
clearly in the 171Å bandpass, we assume that the temperature of the loop corresponds to the peak temperature
of that bandpass. The observed intensity, normalised
with exposure time, is related to the differential emission measure as
Z∞
(3)
I171 = DEM (T ) R171 (T ) dT ,
0
where R171 (T ) is the response function for the 171Å
bandpass. The contribution to the intensity from the
loop itself is determined from fitting at each position
along the loop a Gaussian profile with background. Assuming an isothermal plasma, the DEM becomes trivially
DEM (T ) = n2e LOS δ(T − T0 ) ,
(4)
4
E. Verwichte et al.
Fig. 3.— PFSS potential field extrapolation of active region
NOAA 11283 on 7 Sep 2011 at 00:00 UT. The two oscillating loops
are indicated in red.
where ne is the electron number density, LOS(s) is the
distance across the loop, which takes into account the
geometry of the loop with respect to the line-of-sight direction, and T0 = 0.79 MK is the peak temperature of the
171Å bandpass. Then, the loop electron number density
is found as
s
Iloop (s)
.
(5)
ne (s) =
R171 (T0 ) LOS(s)
Because we assume the temperature corresponds to the
peak temperature of the bandpass (i.e. R171 (T0 ) is maximal), the thus determined density may be regarded as a
lower limit. Panels (c) of Figures 4 and 5 show the density as a function of distance along the loop. We find the
density to vary between 1015 and 1015 m−3 . The profile
is noisy due to line-of-sight interference but a clear trend
of decreasing density towards the loop top can be seen.
We fit an exponential function to the density profile of
the form
z(s)
ne (s) = ne,fp e− H ,
(6)
where z is the height above the photosphere and H is
the density scale-height. We find values for the density
scale-height of 65 Mm and 139 Mm, respectively. If we
assume an isothermal loop, these would correspond to
temperatures of 1.3 MK and 2.8 MK, respectively. The
loops probably do not have these temperatures as they
are not clearly visible in the 193Å and 211Å bandpasses,
whose peak temperature is in this range. So-called superhydrostatic density scale heights have been determined
before (e.g. Van Doorsselaere et al. 2007).
In panels (d) we combine the magnetic field strength
and electron density and find the Alfvén speed as a function of distance along the loop as
VA (s) = p
B(s)
,
µ0 µ̃mp ne (s)
(7)
where µ̃ = 1.2 for coronal abundances. The average
Alfvén speed across the loop, < VA >, is defined as the
Fig. 4.— For loop #1, a comparison of the Alfvén speed derived
seismologically and derived using magnetic extrapolation with potential fields and spectroscopy using SDO/AIA bandpasses. a)
Magnetic field strength from PFSS model at 7 Sep 2011 (00:00
UT), interpolated along the 3d path of the loop. The magnetic
field averages < B > and < BW > are indicated as dashed and
long-dashed lines, respectively. b) AIA 171Åintensity along the
path of the loop. The detailed location of the loop is indicated by
two solid lines. c) Electron number density derived using Eq. (5)
as a function of distance along the loop. The dashed line is a fit
of the form (6). d) Alfvén speed as a function of distance along
the loop. The thick blue curve is the Alfvén speed along the loop
determined using Eq. (7). The horizontal shaded region is the
range of values for the seismologically determined Alfvén speed,
VA,s , using Eq. (2) for an arbitrary value of the loop density contrast. The light shaded region bordered by dotted lines extends
this range by including the measurement errors of loop length and
oscillation period. The thin dashed line is the average Alfvén speed
< VA >. The thick blue dashed line is the Alfvén speed, VA,d .
constant speed that gives the same travel time between
the two foot points (Aschwanden & Schrijver 2011):
−1
 L
ZL
Z
ds
ds
L

.
= ∆t =
⇒ < VA > = L 
VA (s)
< VA >
VA (s)
0
0
(8)
Figures 4 and 5 show that the average Alfvén speed exceeds the seismologically determined Alfvén speed range
by up to a factor of two. This result is consistent with
what was found by Aschwanden & Schrijver (2011). It
seems to indicate that there is a mismatch between seismological and direct methods in determining the loop
Alfvén speed. Following the definition of the average
Alfvén speed we calculate the average magnetic field as
−1
 L
Z
ds

,
(9)
<B > = L 
B(s)
0
Coronal Alfvén speed determination
5
Fig. 6.— Normalised Alfvén speed VA,d /VA,top = ω̃L/A,top as a
function of normalised Alfvén speed scale height Λ/L from solving
Eq. (13). The long-dashed curve corresponds to the solution using
the average Alfvén speed <VA>. The values corresponding to the
two loops are indicated. The inset shows the displacement profile
along the loop. The dashed line is a sine curve.
oscillation period and phase speed (Andries et al. 2009).
In the thin flux-tube limit, the spatial structure of a TLO
may be modelled using the following differential equation
(Dymova & Ruderman 2005, 2006; Verth & Erdélyi 2008)
Fig. 5.— Same as Fig. 4 but for loop #2.
where B(s) is the magnetic field strength along the threedimensional loop path. For the two loops, we find that
the magnetic field strength varies between more than
100 Gauss at the foot points to values around or below
20 Gauss at the loop tops. The average magnetic field
strength is 26 G and 41 G, respectively. These values fall
within the range that was found seismologically with ne
∼ 1015 m−3 .
We may improve the averaging by including a weight
that takes into account the localisation of wave energy
along the loop. Thus, the average Alfvén speed is modified to be
 L
−1
Z
W (s) ds 
< VA,W > = L 
,
(10)
VA (s)
0
with weight function
−1
ZL
.
W (s) = ne (s)ξ 2 (s)L  ne (s)ξ 2 (s)ds

(11)
0
ξ(s) here is the mode displacement profile. When not
using the mode profile from solving Eq. (12), it may be
approximated using ξ(s) = ξ0 sin(πs/L). The average
magnetic field may be weighted in the same manner. As
Table 1 shows, the weighted average Alfvén speeds for
the two loops lies closer to the seismological Alfvén speed
range but is still represents an over-estimation.
4. ROLE OF STRATIFICATION
We wish to take the analysis a step further and consider the role of stratification on the TLO. The longitudinal structuring of the Alfvén speed will modify the
ω2
d2 η(s)
+ 2
η(s) = 0 ,
2
ds
CK (s)
(12)
where η(s) = ξ/a is the transverse loop displacement relative to the local loop radius and ω is the mode frequency.
Instead of modelling the kink speed in Eq. (12) using a
range of valuespof ζ, we take
√ an alternate approach. We
introduce ω̃ = 1 + ζ −1 ω/ 2. We assume that ζ is constant along the loop. Then Eq. (12) is modified to
d2 η
ω̃ 2
+
η = 0,
ds2
VA2 (s)
(13)
From the resulting modified mode frequency, we define a
phase speed, VA,d , as
ω̃L
.
(14)
π
We have two approaches to solve Eq. (13). In the first approach, we solve Eq. (13) completely numerically using
a Runga-Kutta algorithm with adaptive step size (Press
et al. 2007). The mode frequency ω̃ is then found using a
shooting method. A second approach is to first solve Eq.
(13) analytically by modelling the Alfvén speed profile
with an exponential profile of the form VA (s) = VA,top
exp(|s−L/2|/Λ) where Λ represents a typical scale height
of the Alfvén speed. Then, the solution of Eq. (13)
with zero foot point displacement is in terms of Bessel
functions of order 0 (Ferraro & Plumpton 1958; McEwan
et al. 2008)
VA,d =
η(s) = η0 [Y0 (kxL )J0 (kx) − J0 (kxL )Y0 (kx)] ,
(15)
where k = ω̃Λ/VA,top = (Λπ/L)(VA,d /VA,top ), xL =
exp(−L/2Λ) and x = exp(−|s − L/2|/Λ). The wave frequency ω̃, and through Eq. (14) also VA,d , is then found
6
E. Verwichte et al.
as the fundamental mode solution of the dispersion relation
Y0 (kxL )J1 (k) − J0 (kxL )Y1 (k) = 0 ,
(16)
which is solved numerically using e.g. a bracketing rootfinding algorithm. For the two loops using the Alfvén
speed profiles from Figs. 4 and 5 we find fit values of
Λ = 107 Mm and 98 Mm, and VA,top = 1470 kms−1
and 1640 kms−1 , respectively. The numerically derived
values are listed in Table 1.
We have determined VA,d using both numerical and
analytical approaches and they give equivalent results.
Figures 4 and 5 show that for both loops VA,d (found with
the numerical approach) falls inside the observed range
of Alfvén speeds. This suggests that the seismological
and direct methods are consistent.
It also show the importance of interpreting correctly
the observed Alfvén speed range. Our findings indicate
that the average Alfvén speed < VA > is not necessarily
a good measure to compare with the seismology. Figure
6 shows the dependency of ω̃ on Λ and this may be used
to quickly read the solution for given Λ without need of
explicitly solving Eq. (13). The solution is fitted by a
power-law of the form
"
µ ¶−0.93 #
Λ
.
(17)
VA,d = VA,top 1 + 0.27
L
Figure 6 also clearly illustrates the disparity between the
full solution and the use of the average Alfvén speed,
which becomes significant for Λ . L. In the cases studies,
Λ/L ≈ 0.5 and < VA > is nearly 1.5 times as large as VA,d .
5. DETERMINATION OF THE LOOP CROSS-SECTIONAL
PROFILE
If we equate the Alfvén speed VA,s in Eq. (2) with
VA,d , then we can directly calculate the density contrast
ζ, i.e.
µ
¶2
VA,d
−1
ζ
= 2
−1 ,
(18)
Vph
assuming ζ is constant along the loop. For loops #1 and
#2 we find values of ζ −1 = 0.4 and ζ −1 = 0.8, respectively. Note that the uncertainties are large. We may
obtain seismologically more information about the loop
cross-section by including the observed damping rate.
The leading theory that explains the rapid damping of
the oscillations is resonant absorption. It critically depends on the thickness of a thin transition layer, ℓ, over
which the density drops from inner to external conditions. Under the assumption of small thickness, ℓ ≪ a,
it is given by (e.g. Ionson 1978; Hollweg & Yang 1988;
Goossens et al. 1992; Ruderman & Roberts 2002)
ℓ
ζ +1 P
= F
.
a
ζ −1 τ
(19)
where ℓ/a is between 0 and 2. For a half-wavelength sinusoidally varying density profile across the layer, F = 2/π
(Ruderman & Roberts 2002). Equation (19) is strictly
speaking only in the regime where ℓ ≪ a, though it still
provides a relatively accurate extension into the regime
of finite resonance layer widths (Van Doorsselaere et al.
2004). Also, Eq. (19) does not describe any transient
Fig. 7.— Principle how the techniques of coronal seismology,
magnetic extrapolation and spectroscopy and their determination
of the Alfvén speed, magnetic field strength and density are linked
with each other through the formula of the Alfvén speed.
behaviour in the damping (Pascoe et al. 2012). From
the previous Sections we have estimates of all parameters in Eq. (19), except for ℓ/a. We can thus determine
its value. We find ℓ/a = 1.1 for loop #1 and ℓ/a = 1.9
for loop #2. The uncertainties are large and covers the
whole [0,2] interval for loop #2. The uncertainty on ℓ/a
for loop #2 is large due to the large uncertainty in the
damping time.
6. DISCUSSION
We have determined the Alfvén speed in a coronal
loop using two independent methods: a seismological
method applied to a transverse loop oscillation and a
direct method taking the effects of density and magnetic
stratification into account and based on magnetic potential field extrapolation and spectral information from the
AIA bandpasses. We have repeated the study for two oscillating loops seen by AIA/SDO during the same flaring
event, using the 171Å bandpass. For both loops, both
methods give consistent results, which demonstrates that
the technique of coronal seismology produces valid results. We have demonstrated that it is important to correctly interpret the observed Alfvén speed range from
the oscillations. It is not sufficient to compare with an
average Alfvén speed along the loop, which explains the
discrepancy that Aschwanden & Schrijver (2011) initially
found.
In turn, when using a seismologically determined value
of the Alfvén speed, VA,s , as input for modelling or for
comparison with other types of observational data, it is
important to understand what this value means with respect to the Alfvén speed profile of the loop. Equation
(17), where VA,d is replaced by VA,s , provides a method
for connecting the measurement with the Alfvén speed
profile along a longitudinally stratified loop as represented by a loop top value and a scale height.
A main conclusion to draw from this study is that
seismological techniques should not be seen in isolation
Coronal Alfvén speed determination
from other methods. We have shown that through the
formula of the Alfvén speed, the techniques of coronal
seismology, magnetic extrapolation and spectroscopy are
linked. Figure 7 illustrates the connections graphically.
Thus, combining information from multiple techniques
allows to construct more advanced methods of determining physical loop parameters or/and to build in consistency checks. We can envisage several scenarios. The
obvious first scenario is the determination of the loop’s
magnetic field (assuming a longitudinally uniform loop)
where the density from spectroscopy is combined with
the Alfvén speed from seismology (Nakariakov & Ofman
2001). When including the effect of stratification at least
two parameters are required to model the Alfvén speed
profile. To resolve these parameters purely seismologically, it requires a detailed measurement of the spatial
wave displacement profile or of multiple wave harmonics
(Andries et al. 2009). However, when this is not available, magnetic extrapolation and spectroscopy can provide alternate information. We have provided a figure
(Fig. 6) to help find without calculation the expected
Alfvén speed VA,d from the Alfvén speed at the loop top
and the Alfvén speed scale height. Lastly, as demonstrated in Sect. 5, when combining all methods we can
obtain information about the density contrast and transition layer thickness, which were impossible to determine
separately by purely seismological means (Arregui et al.
2008; Goossens et al. 2008).
We have also looked at applying the same comparison
between seismological and direct methods of determining the Alfvén speed to off-limb events but encountered
the difficulty of obtaining an accurate value of the density due to large line-of-sight confusion. Also, the threedimensional path is less precise in that case because it is
established using EUVI/STEREO as the reference (instead of the other way round for on-disk events seen by
AIA/SDO). Furthermore, the magnetic extrapolation is
inaccurate as it is constructed from older or later magnetogram data from near the limb.
We add the following caveats to this study. Firstly,
the magnetic extrapolation is based on a potential field
model that has limitations for active regions, where it is
expected that free magnetic energy is stored in the field,
which ultimately drives eruptions (Wiegelmann & Sakurai 2012). Alignment between the visible loop and the
7
magnetic field can be improved by employing force-free
extrapolation models (e.g. Régnier et al. 2008) instead
and using magnetic maps from HMI/SDO. Secondly, the
determination of the density relied on the observation
from the 171Å bandpass alone. Therefore, the determined density is only a lower limit. Ideally, we wish
to obtain a better constraint on the density by using
multiple bandpasses (Aschwanden et al. 2011; Hannah &
Kontar 2012) and may also incorporate density measurements from at least one loop location using spectrometer data (from e.g. EIS/Hinode, Culhane et al. 2007).
Thirdly, it is critical for the three-dimensional geometry
of the loop to be determined accurately. We have repeated the analysis for different loop inclination angles.
A change of five degrees causes VA,d , derived from extrapolation, to vary between the bottom and top range of the
seismologically determined Alfvén speed range. Therefore, the derived values of loop contrast ζ and transition
layer thickness ℓ, which already have large uncertainties
attached, have to be interpreted with caution.
Finally, we have not included the effect of curvature or
structuring in the external corona (e.g. non-constant ζ
along the loop) that may give rise to lateral wave leakage (Brady & Arber 2005; Verwichte et al. 2006a; Van
Doorsselaere et al. 2009; Pascoe et al. 2009), which modifies the relation between observed phase and kink speed
as well as contributes to the observed oscillation damping. For thin coronal loops this effect is expected to be
secondary.
EV acknowledges financial support from the UK Science and Technology Facilities Council (STFC) on the
CFSA Rolling Grant and the SF fellowship SF/12/004
of the KU Leuven Research Council. TVD acknowledges
funding from the Odysseus programme of the FWOVlaanderen and the EUs Framework Programme 7 as
ERG 276808. CF acknowledges financial support from
STFC under her Advanced Fellowship ST/I003649/1.
RSW acknowledges support of an STFC Ph.D. studentship. AIA data is courtesy of SDO (NASA) and the
AIA consortium. We thank the SOHO, STEREO and
SDO instrument teams for making available data used
in this paper, and Dr Marc de Rosa for useful discussions about the PFSS package.
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